Riding a motorcycle is much more than getting your head working, you need to ensure you're controlling the motorcycle effectively. Body position is key; if you're not comfortable you can't be relaxed and if you're not relaxed you won't be able to use your motorcycle's controls efficiently.
Start by sitting in a relaxed position. This sounds a lot easier than it actually is. Even where you sit on the seat can make a big difference – too far back form the tank and the tendency is to lock out your arms, which impedes steering; further forward with your crotch against the fuel tank tends to straighten your back and has a similar effect by locking out the your elbows. A seating position that encourages a more rounded back and bend arms when holding the handlebars will allow you to be much more precise with your inputs.
The next crucial thing is your hands; a relaxed grip is vital. Gripping the handlebars very tightly can lock out wrists, elbows and shoulders, and effectively prevent you from steering well. You'll also find it impossible to control your throttle, brake well and operate your clutch effectively. Your grip obviously needs to be sufficient to remain in control of the handlebars, but you'll find the force required to do this is surprisingly small. The lighter your grip, the more relaxed your upper body will be. You can lighten your grip on the handlebars by placing less body weight through them. Often this is easier than done. It's more difficult the more sports-focused your motorcycle is – on a big Adventure Bike like a BMW R1200GS it's no problem, but try taking all your upper body weight off the handlebars on a Ducati 1198 traveling at low speeds. In a straight line, with smooth tarmac, it probably doesn't make that much difference in any case, but a relaxed grip will certainly have a big impact on the level of control you have on bumpy roads or while changing direction, cornering or braking.
You can use other muscle groups to help take the weight off your arms.
Try this simple exercise. With your motorcycle on a centerstand or paddock stand, sit on it with you feet on the pegs, lean forward so that your hands are on the handlebars, with a good bend at the elbow. Then take your hands off the handlebars, but hovering about an few centimeters above them. Get a feel for which muscles you're using to maintain this position. Your back? Yes. Your thighs? Quite possible, because you may be squeezing the fuel tank. Stomach muscles? Almost certainly. Knowing which muscle group to bring into play to take your weight off the handlebars will certainly help. (By the way – you should also do this exercise on a motorcycle you're thinking of buying.)
Where you have your feet positioned can also make a difference to how you carry your weight on the motorcycle. It is much easier to take weight off your hands/arms, if you have the balls of your feet on the pegs rather than the instep. It is also much easier to shift your body position and weight...